Aircraft: B737
Phase of flight: Initial Approach
Anomaly: Aircraft Equipment Problem Less Severe

I was acting as a check airman giving [the First Officer] Captain IOE. The takeoff roll was uneventful. During the rotation though; we felt a shaking in the left rear of the aircraft immediately before liftoff. The shaking stopped once airborne. At the time; I thought it felt like the aircraft was 'skipping' to the side due to a little side load from the crosswind. The flight to [the destination] was uneventful until we lowered the gear on the approach. With the gear handle down; we saw the left main was showing a red unsafe light and the gear horn was sounding. I cycled the gear and the same thing happened again. I told [the First Officer] to go around so we could run the checklist. I informed ATC of the problem and began getting vectors. I accomplished the 'Gear Disagree' and 'One or both main landing gear position indicators show unsafe' checklists. With the gear handle down; the main gear indicators showed the red left main light; but the secondary main gear indicator showed green. I retracted the gear and the red light turned off; but the secondary green light stayed illuminated. I lowered the gear again and proceeded to call maintenance control. I spoke with [a maintenance technician] and he said he couldn't think of anything else to do. I said about doing a tower fly-by to get an external look at the gear. [Maintenance] agreed. Before doing the fly-by; I told [the First Officer] I'd feel more comfortable if I took over as pilot flying since he only had one landing in the left seat. He agreed. I told him we would keep the gear down for the remainder of the flight unless we needed to raise it for performance. We performed the fly-by without incident. The tower and an aircraft holding short both said the gear appeared to be down and locked. I decided to still have emergency equipment standing by. I called [maintenance] back and informed him of our plan to land. I also told him to call [operations] so they could come out to pin the gear and tow us in. I informed ATC that we would most likely be stopping on the runway. I told [the First Officer] I wanted to land with the flaps at 40 degrees so we would touch down at the slowest ground speed. I also said that we would keep the auto brakes off so no extra stress was placed on the gear until we were slowed. Our approach speed was 120 with a ref of 115. The touchdown was smooth; but a strong vibration in the back left happened almost immediately. No braking was being applied. A few seconds later the Tower advised us that an aircraft on the taxiway reported seeing one of our left main tires blown out. As we slowed below 70 the vibration disappeared. I added very minor braking to stop us just beyond the runway intersection. I did not want to taxi with a blown out tire; so we shut down the aircraft on the runway. I called operations and they advised that maintenance was on the way. When they arrived they pinned the gear and confirmed the Tire was blown out. Twenty minutes later the tug showed up and pulled us to the ramp. [Report narrative contained no additional information.] B737 flight crew reported that during final approach when they attempted to extend the landing gear they got a 'LANDING GEAR UNSAFE' indication.